rate of change and slope objectives: use the rate of change to solve problems. find the slope of a...
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Rate of Change and Slope
Objectives:
• Use the rate of change to solve problems.
• Find the slope of a line.
The word slope (gradient, incline, pitch) is used to describe the measurement of the steepness of a straight line.
Slope
The slope of a line is also known as the rate of change.
Types of Slopes
Positive Slope m = +
Zero Slope m = 0
Undefined Slope
Negative Slope m = -
Slope is a ratio and can be expressed as:
Slope = Vertical Change or Rise or Horizontal Change Run
2 1
2 1x
y y
x
To find the slope in this lesson you must use….
Slope formula2 1
2 1
y ym
x x
Vertical change = 14
P2(x2, y2)
P1(x1, y1)
Horizontal change = 7
Slope = vertical change
horizontal change
Slope = 14 or 2
7
Is the slope positive or negative?
Positive
Find the slope of each line:
run
rise
2
3 run
rise
3
5
Practice
Find the slope of the line that passes through (-1, 4) and (1, -2).Then graph the line.
2 1
2 1x
ym
y
x
1 ( )
2
1
4m
6
23m
Find the slope of the line that passes through each pair of points.
1. (7, 6), (7, 4) 2. (9, 3), (7, 2)
= undefined slope = ½
12
12
xx
yym
12
12
xx
yym
77
64
m
0
2m
97
32
m
2
1
m
Find the slope of the line that passes through each pair of points.
3. (1, 2) (-1, 2) 4. (9, -4) (7, -1)
slope = 0 slope =23
Graph on the coordinate plane m= -3 Passes through
(-1, 2)
Graph on the coordinate plane m= 1/2 Passes through
(2, 3)
Graph on the coordinate plane m= undefined Passes through
(3, 1)
Graphing Equations
● Example: Graph the equation -5x + y = 2Solve for y first.
-5x + y = 2 Add 5x to both sides y = 5x + 2
● The equation y = 5x + 2 is in slope-intercept form, y = mx+b. The y-intercept is 2 and the slope is 5. Graph the line on the coordinate plane.
x
y
Graph y = 5x + 2
Graphing Equations
Graph 4x - 3y = 12
● Solve for y first
4x - 3y =12 Subtract 4x from both sides
-3y = -4x + 12 Divide by -3
y = x + Simplify
y = x – 4● The equation y = x - 4 is in slope-intercept form,
y=mx+b. The y -intercept is -4 and the slope is . Graph the line on the coordinate plane.
Graphing Equations
12-3
43
43
43
-4-3
Graph y = x - 4
x
y
43
Graphing Equations
Find the value of r so that the line through (r, 6) (10, -3) has a slope of 2
3
12
12
xx
yym
r
10
9
2
3
r
10
63
2
3
-3(10 – r) = 2(-9)
-30 + 3r = -18
3r = 12
r = 4
Slope Formula
Let (r, 6) = (x1, y1) and (10, -3) = (x2, y2)
Subtract
Cross Multiply
Use the Distributive Property and simplify
Add 30 to each side and simplify
Divide each side by 3 and simplify
The line goes through (4, 6)
Find the value of r so the line that passes through each pair of points has the given slope.
1. (1, 4) (-1, r); m = 2
2. (r, -6) (5, -8); m = -8
r = 0
r = 4.75
Questions . . . ?
Comments . . . ?
Concerns . . . ?